(Modeling) Length of a Sag Curve When a highway goes downhill and then uph...

(Modeling) Length of a Sag Curve When a highway goes downhill and then uphill, it is said to have a sag curve. Sag curves are designed so that at night, headlights shine sufficiently far down the road to allow a safe stopping distance. See the figure.

The minimum length L of a sag curve is determined by the height h of the car’s headlights above the pavement, the downhill gradeθ₁ < 0° , the uphill grade θ₂ > 0° , and the safe stopping distance S for a given speed limit. In addition, L is dependent on the vertical alignment of the headlights. Headlights are usually pointed upward at a slight angle a above the horizontal of the car. Using these quantities, for a 55 mph speed limit, L can be modeled by the formula

where S 6 L. (Source: Mannering, F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.)

a) Compute L if h = 1.9 ft, α = 0.9° , θ₁ = -3° ,θ₂ = 4° , and S = 336 ft.

(b) Repeat part (a) withα= 1.5° .

(c) How does the alignment of the headlights affect the value of L?